Some remarks on the bias of the MPN method

International Journal of Food Microbiology, 13 (1991) 131-142 © 1991 Elsevier Science Publishers B.V. 0168-1605/91/$03.50

131

FOOD 00407

Some remarks on the bias of the M P N m e t h o d Oliver R e i c h a r t Department of Microbiology and Biotechnology, Unioersity of Horticulture and Food lndustO,, Budapest, Hungary (Received 20 February 1990; accepted 28 January. 1991)

The estimation of cell number by using the MPN method was studied mathematically for three parallel inoculations from three decimal dilutions. As a result of computation it can be established that these estimations at a range of microbial numbers over 10 are biased. Confidential estimation could be obtained at cell numbers betw~n 0 and 10 accepting only a few combinations of positive and negative test tubes. These are 1 0 0, 2 0 0, 3 0 0, 3 1 0 and 3 2 0. Key words: MPN method; Biased estimation; Probability; Relative frequency

Introduction Nowadays the microbiological standards for foods are becoming increasingly severe, so the MPN method will probably often replace plate counting. MPN values and confidence intervals can be calculated for any combination of positive and negative test tubes, but the probability of many combinations is so low that they will practically never be obtained. Leaving out the improbable values, de Man (1975) restricted the MPN tables to results having realistic probability. He suggested two categories. Category 1, normal results, obtained in 95% of cases; Category 2, less likelyresults, obtained only in 4% of cases. Results that are even less likely than those of category 2 are always unacceptable. This results are often due to contamination of sterile tubes. MPN theory and tables with confidence limits calculated by Man (1983) are critically reviewed by Jarvis (1989). It seems, that the problem of the probability of Most Probable Number has been solved. However, there is an other, neglected problem: estimation of microbial numbers by using MPN method is biased in some cases. The basic question is the following: If the real microbial number is N, what is the most probable combination of the positive and negative tubes? So the problem of the most probable numbers is reduced to that of the most probable combination of test tube results. Correspondence address: O. Reichart, Department of Microbiology and Biotechnology, University of Horticulture and Food Industry, Sornloi ut 14-16, H - I l l 8 Budapest, Hungary.

132 Present work offers a simple solution of this problem calculating the mean values of the estimated MPNs as a function of real cell numbers.

Calculation

method

The interval of consistent estimation of microbial numbers was investigated in the most frequently used case, i.e. three parallel inoculations from three decimal dilutions. Calculations were made by a P C / A T BIOS computer. Calculation method can be described as follows. Probability distribution The probability of a negative (sterile) test tube in the ith dilution: n! ).-,, P' = s~i( n - s,)! "PSi" (1 - P i

(1)

where: Pi, probability of s i sterile tubes; n: number of inoculated tubes; s i, number of sterile tubes; p~, probability of negative (sterile) tube; Pi = e-X~, where Ai is the number of microbes per inoculum. In the case of three decimal dilutions: hI

= N

Pl =

e-N

A2 = 0.1 N

p2 = e - ° a ~

A3 = 0.01 N

P3 = e-°'m N

Inoculating three parallel test tubes from every dilution, the probability of sterile tubes can be calculated as follows.

el

3~ =

s~!(3 - sl)!

-p~,. (I _ pI)3-,,

3! . p ? . (1 - p~)~-': s2!(3 ~ $2 ~ ~ 3~ P3 = .pj3. (1 --p3)3-s3

r~ =

$3!(3 S3)!

The probability of the combination of s~, s 2, s 3 sterile tubes

P = P, . P2. P3

(2)

Calculating the P probability at any number of microbes the probability distribution curves can be obtained at any sl, s 2, s 3 combinations (except 3, 3, 3 and 0, 0, 0).

133 TABLE I M P N table for three decimal dilutions with three parallel inoculations (P,,u~ is m a x i m u m value of probabilities) N u m b e r of positive results

Pm~

MPN

000 00 1 002 003 0 1 0 ** 0 11 012 0 13 020 02 1

0.0033 1.5 × 10 - s 3.7 x 10 - 8 0.0336 4.5 X 10 - 4 3.5×10 -6 1.2×10 -s 0.0016 3.6× 10 - s

< 0.3 0.3 0.6 0.9 0.3 0.61 0.92 1.2 0.62 0.93

0 22

3.9 × 10 -~

1.2

222

5.4 x 10- 5

3.5

0 0 0 0

1.8 X 10 - 9 4.2 x 10 - s 1.4 x 10 - 6 2.0x I0 -s

1.6 0.94 1.3 1.6

2 2 2 2

7.0 x 1 0 - ~ 0.0022 2.1 x 10 - 4

4.2 2.9 3.6

0 33 I00 *

1.2xlO -I° 0.3920

1.9 0.36

233 300 *

8.6 x 1 0 -6 1.4xlO -~ 0.3410

4.4 5.3 2.3

10 1 ** 10 2 10 3 110 * 111 112 113 1 2 0 ** 12 1 122 12 3 130 13 1 132 133

0.0062 5.6× 10 - 5 2.4 × 1 0 - ~ 0.0645 0.0018 2.4× 10 - s 1 . 4 × 1 0 -7 0.0063 2.5 x 10 - 4 4 . 4 × 10 - 6 3.2 × 10 - s 3.0 × 10 - 4 1.6 X 10 - s 3.6× 10 -~ 3.2X10 - 9

0.72 1.1 1.5 0.73 1.1 1.5 1.9 1.1 1.5 2.0 2.4 1.6 2.0 2.4 2.9

30 1* 302 303 3 10 * 311* 312 313 320 * 321* 3 2 2 ** 323 330 * 331* 332 * 333

0.0310 0.0016 4.3 × 1 0 - 5 0.3743 0.0658 0.0065 3.2×10 -4 0.3282 0.1251 0.0247 0.0024 0.3659 0.4277 0.A.A.A:.

3.9 6.4 9.5 4.3 7.5 12 16 9.3 15 21 29 24 46 110 >110

2 3 3 3

3 0 1 2

N u m b e r of positive results 200 20 1 202 20 3 2 10 211 212 213 220 221 2 3 3 3

Pmax

* **

MPN

0.3193 0.0112 1.9 x 10 - 4 1.4 x 10 - 6 0.1196 0.0063 1.5X10 - 4 1.5×10 -6 0.0232 0.0017

* **

*

3 0 1 2

0.91 1.4 2.0 2.6 1.5 2.0 2.7 3.4 2.1 2.8

* Category 1: obtained in 9 5 $ of cases. * * Category 2: obtained only in 4% of cases.

According

to

the

(k = n - s) are related

convention

in

MPN

to the cell numbers

tables

the

numbers

corresponding

the probability curve. Because the maximum values of probabilities i n l i t e r a t u r e t h e s e v a l u e s a r e s u m m a r i z e d i n T a b l e I. The probability

curves belonging

t u b e s c a n b e s e e n i n F i g . 1. T h e maximum over 0.05.

to the most

figure shows

of

positive

to the maximum

tubes

value of

are not available

probable

combinations

probability

histograms

of positive having

their

134 P 3,~

05.

0 4,,

33, ~o

~3o

c . . . ~ ~

IY/\~-

10 20 30 40 N Fig. 1. Probability curves (Probability ( P ) versus cell n u m b e r s ( N ) ) corresponding to the combinations of positive test tubes (three decimal dilutions with three parallel inoculations).

On the base of probability curves the problem of most probable combination of positive results can be interpreted. Let us assume that the number of microbes in the first dilution is 15. In this case the most probable score of positive tubes is 3, 3, 0 instead of 3, 2, 1 although MPN3.2, t = 15. The probable results in the order of their probability are as follows: MPN3,3. 0 = 24 MPN3.2, 0 = 9.3 MPN3.3.1 -- 46 MPN3.2.1 = 15 MPN3.1. 0 = 4.3 MPN3,1A = 7.5 The mean of cell numbers estimated by M P N method can be calculated as follows:

= E Pi (MPN)i

E P,

(3)

where: N, mean of estimated n u m b e r of microbes; Pi, probability of k t, k2, k 3 combination belonging to the real cell number ( N ) ; ( M P N ) i, M P N belonging to the k 1, k 2, k 3 combination. By introducing relative frequency

Pi fi= 2 p i

(4)

the (3) equation can be calculated: = y ' f i (MPN),

(5)

135 TABLE

II

Calculation of the m e a n of estimated M P N s

( N =15, Pk = 0.01)

Combination of positive tubes

MPN

P

P f = ~----:

330 320 331 321 310 311 332 322

24 9.3 46 15 4.3 7.5 110 21

0.2990 0.2576 0.1452 0.1251 0.0740 0.0359 0.0235 0.0202

0.3049 0.2628 0.1480 0.1276 0.0"/55 0.0366 0.0240 0.0206

Y'-P = 0.9805

1.0000

r2.

fi = Eli (MPN)~ = 22.16 For explanation of symbols see section on calculation method.

Using the fi relative frequency, information can be obtained about the reality of a k~, k 2, k 3 combination. Assuming that the real cell number N - - 1 5 and the probability threshold Pk ffi 0.01 (every kl, k2, k 3 combination having probability less than 0.01 at N = 15 is neglected), the calculated mean of MPNs IV = 22.16. (The results of calculation are given in Table II.)

i; 70

•O

xX

xxxx~ 5o

1¢

xXX" x /

xX~

,~,N

x P~: oo~ OP~ =025

;o

~o

~

~o

~o

Fig. 2. Estimated mean of MPNs ( N ) as a function of real cell number ( N ) . Pk = probability threshold of obtaining a combination.

136 Results and Discussion Fig. 2 shows different

results

probability

obtained

thresholds.

by calculating Comparing

the

the mean estimated

of estimated cell

number

MPNs with

T A B L E III Estimated cell n u m b e r s at different probability thresholds (Pk) and real cell n u m b e r s ( N ) N

Pk 0.30

0.25

0.20

0.15

0.10

0.05

0.01

1 2 3 4 5 6 7 8 9 10

0.9 2.3 3.3 4.3 4.3 4.3 6.8 9.3 9.3 9.3

0.9 2.3 3.3 3.5 4.3 6.6 6.8 7.1 9.3 9.3

0.9 3.1 3.3 3.5 6.3 6.6 6.8 7.1 7.3 9.3

1.1 2.6 3.3 4.8 5.3 6.6 6.8 7.1 11.0 11.9

1.2 2.5 4.2 4.8 5.3 5.8 9.3 10.2 11.0 12.3

1.4 2.5 3.9 4.8 6.5 7.9 8.9 9.8 10.7 14.1

1.3 2.8 4.4 5.6 6.8 8.3 9.6 10.9 12.2 13.6

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

9.3 9.3 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 24.0 34.1 34.4 34.6 34.8 35.1 35.3 35.5 35.7 35.9 36.1 36.3 36.4 36.6 46.0 46.0 46.0

9.3 9.3 16.2 16.7 17.2 24.0 24.0 24.0 24.0 24.0 33.1 33.3 33.6 33.9 34.1 34.4 34.6 34.8 35.1 35.3 35.5 35.7 35.9 36.1 36.3 36.4 36.6 36.8 36.9 37.1

15.2 15.7 16.2 16.7 17.2 17.7 18.1 26.0 32.5 32.8 33.1 33.3 33.6 33.9 34.1 34.4 34.6 34.8 35.1 35.3 35.5 35.7 35.9 36.1 36.3 36.4 36.6 36.8 37.0 53.6

12.7 15.7 16.2 16.7 17.2 24.1 25.1 26.0 26.8 27.6 28.4 33.3 33.6 33.9 34.1 34.4 34.6 34.8 35.1 35.3 35.5 35.7 48.8 50.0 50.2 50.9 51.6 52.3 53.0 53.6

13.0 13.7 18.2 21.0 21.9 22.8 23.6 24.4 25.4 26.0 26.8 27.5 29.8 30.4 34.1 34.4 44.4 45.1 45.9 46.6 47.4 48.1 48.8 50.0 50.2 50.9 51.6 52.3 53.0 53.6

15.3 16.5 18.2 19.4 20.5 21.5 23.6 24.4 25.2 30.8 32.0 33.3 34.5 35.8 37.0 38.1 39.3 40.4 41.5 42.6 45.5 48.1 48.8 50.0 50.2 50.9 51.6 52.3 53.0 53.6

15.0 17.4 18.9 20.7 22.2 23.6 25.1 26.6 28.0 29.4 30.8 32.2 33.5 35.0 36.6 37.7 38.9 40.0 41.1 42.1 43.1 44.2 45.2 46.2 47.1 48.1 49.0 49.9 50.7 51.6

at the

137

~I.-N 12

0

0

J

~

2

0

o

i

L

4

0

(320)

o t310)

i,

~ ' ~ ' ;o ,iN Fig. 3. The MPNs corresponding to the most probable combination of positive test tubes. (N. estimated means of MPNs; N, real cell number).

theoretical N = N values a great difference can be found. The estimated values are systematically greater than the real ones. Consistent estimation can be obtained only at cell numbers below 10 in the case of a probability threshold of 0.25.

X

/

0

0

0

n

o

o//d/ o o

0 Pk :025 X P= : 0 1 0 f i 2

I 4

,

!

6

,

i 8

! 10

i

l 12

N

Fig. 4. Estimated mean of MPNs ( N ) as a function of real cell number ( N ) corresponding to different probability thresholds (Pk).

138 T h e e s t i m a t e d cell n u m b e r s ( N ) a r e s u m m a r i z e d i n T a b l e III. A t cell n u m b e r s b e l o w 10 t h e r e is n o s y s t e m a t i c d i f f e r e n c e b e t w e e n t h e real a n d e s t i m a t e d v a l u e s . Fig. 3 s h o w s t h e m o s t p r o b a b l e r e s u l t s ( M P N s c o r r e s p o n d i n g t o t h e m o s t probable combination of positive and negative tubes) when the MPN out only once.

t e s t is c a r r i e d

TABLE IV Relative frequency ( f ) of MPN results as a function of the real cell number (N). Probability threshold: Pk = 0.10. Standard deviation, SD N

Probability

f

1

200 10 0 300 210

Results

MPN 0.91 0.36 2.3 1.5

0.3170 0.1845 0.1816 0.1000

0.4048 0.2356 0.2319 0.1277

Estimated N ( N ) 1.18

2

300 310 200 210

2.3 4.3 0.9 1.5

0.3341 0.2219 0.1569 0.1042

0.4089 0.2716 0.1920 0.1275

2.47

3

3 10 300 320

4.3 2.3 9.3

0.3346 0.3198 0.1171

0.4343 0.4138 0.1519

4.23

4

3 10 30 0 320

4.3 2.3 9.3

0.3729 0.2527 0.1834

0.46090.3124 0.2267

4.81

5

3 10 320 300

4.3 9.3 2.3

0.3663 0.2376 0.1882

0.4624 0.3000 0.2376

5.32

6

310 320 300

4.3 9.3 2.3

0.3380 0.2779 0.1370

0.4489 0.3691 0.1820

5.78

7

320 3 10 330

9.3 4.3 24.0

0.3052 0.3011 0.1031

0.4302 0.4244 0.1454

9.32

8

320 3 10 3 30

9.3 4.3 24.0

0.3212 0.2621 0.1312

0.4495 0.3668 0.1836

10.17

9

320 3 10 330

9.3 4.3 24.0

0.3278 0.2246 0.1595

0.4605 0.3155 0.2240

11.02

10

320 3 10 330 32 1

9.3 4.3 24.0 15.0

0.3266 0.1901 0.1871 0.1031

0.4048 0.2356 0.2319 0.1277

12.26

SD =1.56

139 TABLE V

Relative frequency ( f ) of MPN results as a function of the real cell number (N). Probability threshold: Pk = 0.15. Standard deviation, SD. N

Results

MPN

Probability

f

Estimated N ( N )

1

200 100 300

0.91 0.36 2.3

0.3170 0.1845 0.1816

0.4641 0.2701 0.2658

1.13

2

300 3 10 200

2.3 4.3 0.91

0.3341 0.2219 0.1569

0.4687 0.3113 0.2201

2.62

3

3 I 0 300

4.3 2.3

0.3346 0.3188

0.5121 0.4879

3.32

4

3 10 300 320

4.3 2.3 9.3

0.3729 0.2527 0.1834

0.4609 0.3124 0.2267

4.81

5

3 10 320 300

4.3 9.3 2.3

0.3663 0.2376 0.1882

0.4624 0.3000 0.2376

5.33

6

310 320

4.3 9.3

0.3380 0.2779

0.5488 0.4512

6.56

7

320 310

9.3 4.3

0.3052 0.3011

0.5034 0.4966

6.82

8

320 3 10

9.3 4.3

0.3212 0.2621

0.5507 0.4493

7.05

9

320 3 10 330

9.3 4.3 24.0

0.3278 0.2246 0.1595

0.4605 0.3155 0.2240

11.02

10

320 3 10 3 30

9.3 4.3 24.0

0.3266 0.1901 0.1871

0.4641 0.2701 0.2658

11.86

SD ~ 1.06

When the test is performed many times and results of category 2 are accepted, too, the reliability of the estimation decreases (Fig. 4). Tables IV-VIII summarize results obtained by calculating the estimated mean of MPNs at different probability thresholds and microbial numbers below 10. In the Tables beside the N values and relative frequency of the combinations the standard deviations are given as well. Standard deviation (SD) was calculated as follows: )2

10

E()9- N SD =

1

9

(6)

140 TABLE VI Relative frequency ( f ) of MPN results as a function of the real cell number (N). Probability threshold: Pk = 0.20. Standard deviation: SD N

Results

MPN

Probability

f

Estimated N ( N )

1

200

0.91

0.3170

1.0000

0.91

2

300 3 10

2.3 4.3

0.3341 0.2219

0.6009 0.3991

3.10

3

310 30 0

4.3 2.3

0.3346 0.3188

0.5121 0.4879

3.32

4

3 10 300

4.3 2.3

0.3729 0.2527

0.5960 0.4040

3.49

5

3 10 320

4.3 9.3

0.3663 0.2376

0.6065 0.3935

6.27

6

3 10 320

4.3 9.3

0.3380 0.2779

0.5488 0.4512

6.56

7

320 3 10

9.3 4.3

0.3052 0.3010

0.5034 0.4966

6.82

8

320 3 10

9.3 4.3

0.3212 0.2621

0.5507 0.4493

7.05

9

320 3 10

9.3 4.3

0.3278 0.2246

0.5934 0.4066

7.27

10

320

9.3

0.3266

1.0000

9.3 SD ~ 0.94

TABLE VII Relative frequency ( f ) of MPN results as a function of the real cell number ( N ) . Probability threshold: Pk = 0.25. Standard deviation: SD N 1 2 3 4 5 6 7 8 9 10

Results

MPN

Probability

f

Estimated N ( N )

200 300 3 10 300 310 300 3 10 3 10 3 20 320 3 10 3 20 3 10 320 320

0.91 2.3 4.3 2.3 4.3 2.3 4.3 4.3 9.3 9.3 4.3 9.3 4.3 9.3 9.3

0.3170 0.3341 0.3346 0.3188 0.3729 0.2527 0.3663 0.3380 0.2779 0.3052 0.3011 0.3212 0.2621 0.3278 0.3266

1.0000 1.0000 0.5121 0.4879 0.5960 0.4040 1.0000 0.5488 0.4512 0.5034 0.4966 0.5507 0.4493 1.0000 1.0000

0.91 2.3 3.32 3.49 4.3 6.56 6.82 7.05 9.3 9.3 SD ~ 0.60

141 TABLE VIII Relative frequency ( f ) of MPN results as a function of the real cell number (N). Probability threshold: Pk = 0.30. Standard deviation (SD) N

Results

MPN

Probability

f

Estimated N ( N )

1

20 0

0.91

0.3170

1.0000

0.91

2

300

2.3

0.3341

1.0000

2.3

3

3 10 300

4.3 2.3

0.3346 0.3188

0.5121 0.4879

3.32

4

310

4.3

0.3729

1.001~

4.3

5

3 10

4.3

0.3663

1.0000

4.3

6

3 10

4.3

0.3380

1.0000

4.3

7

320 3 10

9.3 4.3

0.3052 0.3011

0.5034 0.4966

6.82

8

3 20

9.3

0.3212

1.0000

9.3

9

320

9.3

0.3278

1.0000

9.3

10

320

9.3

0.3266

1.0000

9.3 SD = 0.814

W h e n c o m p a r i n g SD values c o r r e s p o n d i n g to different p r o b a b i l i t y thresholds a m i n i m u m value at Pk = 0.25 can be established. I n this case o n l y the following c o m b i n a t i o n of positive tubes can be o b t a i n e d : 2 0 0, 3 0 0, 3 1 0, 3 2 0. Beside these c o m b i n a t i o n s , taking into a c c o u n t other p r o b a b l e c o m b i n a t i o n s (see T a b l e s I V - V I I I ) the acceptable results a r e : 1 0 0 , 1 1 0 , 2 0 0 , 2 1 0 , 3 0 0 , 3 1 0 , 3 1 1 , 3 2 0 , 3 2 1 * 3 3 1 *. The results m a r k e d with * are less likely t h a n others, so their a c c e p t a n c e m a y result in an erroneous estimation. F r o m the calculations the following c o n c l u s i o n s can be drawn. (1) T h e M P N m e t h o d results in biased e s t i m a t i o n of cell n u m b e r over 10. (2) M i c r o b i a l n u m b e r s below 10 could be estimated by accepting only a few c o m b i n a t i o n s of positive a n d negative test tubes. These are the following; 1 0 0, 2 0 0, 3 0 0, 3 1 0 a n d 3 2 0. By accepting only these c o m b i n a t i o n s the m e a n difference b e t w e e n the estimated a n d real microbial n u m b e r does not exceed 1 cell n u m b e r .

References De Man. J.C. (1975) The probability of Most Probable Numbers. Eur. J. Appl. Microbiol. 1.67-68. De Man. J.C. (1983) MPN tables, Corrected. Eur. J. Appl. Microbiol. Biotechnol. 17, 301-305. Jarvis, B. (1989) Statistical Aspects of the Microbiological Analysis of Foods, pp. 117-141. Elsevier, Amsterdam.